Properties

Label 29.6.4.1
Base \(\Q_{29}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $S_3$ (as 6T2)

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Defining polynomial

\(x^{6} + 232 x^{3} + 22707\)  Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $6$
Ramification exponent $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{29}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 29 })|$: $6$
This field is Galois over $\Q_{29}.$

Intermediate fields

$\Q_{29}(\sqrt{2})$, 29.3.2.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{29}(\sqrt{2})$ $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} - 29 t^{3} \)$\ \in\Q_{29}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$S_3$ (as 6T2)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed